Superradiant instabilities in astrophysical systems

Light bosonic degrees of freedom have become a serious candidate for dark matter, which seems to pervade our entire Universe. The evolution of these fields around curved spacetimes is poorly understood but is expected to display interesting effects. In particular, the interaction of light bosonic fields with supermassive black holes, key players in most galaxies, could provide colorful examples of superradiance and nonlinear bosenovalike collapse. In turn, the observation of spinning black holes is expected to impose stringent bounds on the mass of putative massive bosonic fields in our Universe. Our purpose here is to present a comprehensive study of the evolution of linearized massive scalar and vector fields in the vicinities of rotating black holes. The evolution of generic initial data has a very rich structure, depending on the mass of the field and of the black hole. Quasinormal ringdown or exponential decay followed by a power-law tail at very late times is a generic feature of massless fields at intermediate times. Massive fields generically show a transition to power-law tails early on. For a certain boson field mass range, the field can become trapped in a potential barrier outside the horizon and transition to a bound state. Because there are a number of such quasibound states, the generic outcome is an amplitude modulated sinusoidal, or beating, signal, whose envelope is well described by the two lowest overtones. We believe that the appearance of such beatings has gone unnoticed in the past, and in fact mistaken for exponential growth. The amplitude modulation of the signal depends strongly on the relative excitation of the overtones, which in turn is strongly tied to the bound state geography. A fine-tuning of the initial data allows one to see the evolution of the nearly pure bound state mode that turns unstable for sufficiently large black hole (BH) rotation. For the first time we explore massive vector fields in a generic black hole background that are difficult, if not impossible, to separate in the Kerr background. Our results show that spinning BHs are generically strongly unstable against massive vector fields.

Figure 1

Time evolution of a dipole ($l=1$, $m=0$) scalar Gaussian wave packet in Schwarzschild background. We clearly observe the main features of such a field: (i) a prompt response at early times followed by (ii) the quasinormal mode ringdown and (iii) a late-time tail.

Figure 2

Amplitude of the scalar field extracted at $r_{\mathrm{ex}}=10M$ as function of time obtained from the evolution of a scalar field initialized as spherical shell with space-dependent mass ${\mu }_S^2=-10M^2/r^4$ around a nonrotating BH.

Figure 3

Evolution of a Gaussian profile of a massless scalar field with width $w=2M$ centered at $r_0=12M$ around a Schwarzschild BH (left) and around a Kerr BH with $a/M=0.99$ (right). We depict the $l=0$ (solid black line) and $l=m=1$ (red dashed line) multipoles.

Figure 4

Left and center: Initial profile of $|{\Psi }_{11}{|}^2$ for a bound state $m=1$ dipole configuration with $M{\mu }_S=0.42$ in a Schwarzschild (left) and $a/M=0.99$ Kerr background (middle). Black solid lines represent the fundamental mode and red dashed lines the first overtone. Right: Relative change of the modulus of the (1,1) mode extracted at $r_{\mathrm{ex}}=20M$.

Figure 5

Upper left: $l=m=1$ multipole of run sS_m001 extracted at $10M$. The solid (black) line refers to numerical data, the dashed (red) to an oscillatory tail fit. Upper right: Same for sS_m100, extracted at $r_{\mathrm{ex}}=20M$ (black solid line) together with the tail fit (red dashed line, we omit the oscillatory term for clarity). Lower panels: $l=m=0$ multipole of runs sS_m042 (left, extracted at $20M$) and sK_m042 (right, extracted at $25M$). The solid black lines refer to the numerical data while the red dashed lines denote the fit to the envelope of the oscillatory late-time tail.

Figure 6

Real part of the $m=1$ dipole obtained at selected extraction radii $r_{\mathrm{ex}}$ for a massive scalar field with $M{\mu }_S=0.42$ in a Schwarzschild (top panels) and Kerr background with $a=0.99M$ (bottom panels). The location of the node of the first overtone is $r_{\mathrm{ex}}=22.5M$ for the Schwarzschild case (upper left-most panel) and $r_{\mathrm{ex}}=26.5M$ for the Kerr case (bottom center panel). $r_{\mathrm{ex}}\sim 50M$ corresponds to the overtone’s local maximum in the Kerr case.

Figure 7

Spectra of the $l=m=1$ mode of the massive scalar field with $M{\mu }_S=0.42$ evolved in the background of a Schwarzschild (left) or Kerr BH with $a/M=0.99$ (right). The lines correspond to the waveforms measured at different extraction radii. In particular, $r_{\mathrm{ex}}=22.5M$ (left) and $r_{\mathrm{ex}}=26.5M$ (right) correspond to the nodes of the first overtone.

Figure 8

Bottom panels: Convergence plot of the $l=m=0$ (left) and $l=m=1$ (right) modes of simulation sK_042. We present the differences between the coarse-medium and medium-high resolution runs, where the latter is amplified by $Q_2=1.66$ indicating second-order convergence.

Figure 9

The $l=1$, $m=0$ multipole of ${\Phi }_2$, extracted at $r_{\mathrm{ex}}=10M$ from the time evolution of a Gaussian pulse of width $w=2M$ (black solid line) and $w=30M$ (red dashed line) around a Schwarzschild BH.

Figure 10

Time evolution of the $l=m=1$ mode of ${\Phi }_2$ of run v1K2_m000, i.e., a massless vector field in a Kerr background with $a/M=0.99$ extracted at $r_{\mathrm{ex}}=10M$.

Figure 11

Time evolution of the Proca field with mass coupling $M{\mu }_V=0.1$ (black solid line) and $M{\mu }_V=0.2$ (red dashed line) in a Schwarzschild background. We show the $l=1$, $m=0$ multipole of the Newman-Penrose scalar ${\Phi }_2$, extracted at $r_{\mathrm{ex}}=10M$.

Figure 12

Time evolution of the Proca field with mass coupling $M{\mu }_V=0.40$ in Kerr background with $a/M=0.99$, at different extraction radii. We plot the $l=m=1$ mode of the Newman-Penrose scalar ${\Phi }_2$ (top panels) and of the scalar component $\phi$ (bottom panels).

Figure 13

Spectra of the $l=m=1$ mode of the Newman-Penrose scalar ${\Phi }_2$ (left) and of the scalar component $\phi$ (right) excited by the Proca field with mass coupling $M{\mu }_V=0.40$ in Kerr background with $a/M=0.99$. Different lines correspond to the waveforms measured at different extraction radii.